2.1 Daily Mortality data

We import daily data series for death counts (which are not freely available) and temperatures. This dataset is constructed by merging:

• daily mortality data,
• daily temperature data.

daily_data <- get(load(file =  paste0(fold_data, "/daily_data.RData")))

# Summary of the data - list with 2 items for females (`f`) and males (`m`)
head(daily_data)

## $m
##           datedec age_bk nb_deaths years     tmax    tmax3        tmin
##     1: 1980-01-01   0-64       265  1980 5.842857 6.464286  0.22142857
##     2: 1980-01-01  65-74       190  1980 5.842857 6.464286  0.22142857
##     3: 1980-01-01  75-84       246  1980 5.842857 6.464286  0.22142857
##     4: 1980-01-01    85+       125  1980 5.842857 6.464286  0.22142857
##     5: 1980-01-02   0-64       276  1980 3.742857 5.173810 -0.95000000
##    ---                                                                
## 58436: 2019-12-30    85+       324  2019 8.085714 8.195238  0.02142857
## 58437: 2019-12-31   0-64       145  2019 7.285714 7.278571  0.22142857
## 58438: 2019-12-31  65-74       175  2019 7.285714 7.278571  0.22142857
## 58439: 2019-12-31  75-84       215  2019 7.285714 7.278571  0.22142857
## 58440: 2019-12-31    85+       313  2019 7.285714 7.278571  0.22142857
##            tmin3     tavg    tavg3  time dow month
##     1: 1.8190476 2.289286 3.801190     1   3     1
##     2: 1.8190476 2.289286 3.801190     1   3     1
##     3: 1.8190476 2.289286 3.801190     1   3     1
##     4: 1.8190476 2.289286 3.801190     1   3     1
##     5: 0.2261905 1.550000 2.483333     2   4     1
##    ---                                            
## 58436: 2.0380952 3.271429 4.640476 14609   2    12
## 58437: 0.3928571 3.464286 3.378571 14610   3    12
## 58438: 0.3928571 3.464286 3.378571 14610   3    12
## 58439: 0.3928571 3.464286 3.378571 14610   3    12
## 58440: 0.3928571 3.464286 3.378571 14610   3    12
## 
## $f
##           datedec age_bk nb_deaths years     tmax    tmax3        tmin
##     1: 1980-01-01   0-64       122  1980 5.842857 6.464286  0.22142857
##     2: 1980-01-01  65-74       117  1980 5.842857 6.464286  0.22142857
##     3: 1980-01-01  75-84       280  1980 5.842857 6.464286  0.22142857
##     4: 1980-01-01    85+       271  1980 5.842857 6.464286  0.22142857
##     5: 1980-01-02   0-64       116  1980 3.742857 5.173810 -0.95000000
##    ---                                                                
## 58436: 2019-12-30    85+       552  2019 8.085714 8.195238  0.02142857
## 58437: 2019-12-31   0-64       107  2019 7.285714 7.278571  0.22142857
## 58438: 2019-12-31  65-74        98  2019 7.285714 7.278571  0.22142857
## 58439: 2019-12-31  75-84       181  2019 7.285714 7.278571  0.22142857
## 58440: 2019-12-31    85+       514  2019 7.285714 7.278571  0.22142857
##            tmin3     tavg    tavg3  time dow month
##     1: 1.8190476 2.289286 3.801190     1   3     1
##     2: 1.8190476 2.289286 3.801190     1   3     1
##     3: 1.8190476 2.289286 3.801190     1   3     1
##     4: 1.8190476 2.289286 3.801190     1   3     1
##     5: 0.2261905 1.550000 2.483333     2   4     1
##    ---                                            
## 58436: 2.0380952 3.271429 4.640476 14609   2    12
## 58437: 0.3928571 3.464286 3.378571 14610   3    12
## 58438: 0.3928571 3.464286 3.378571 14610   3    12
## 58439: 0.3928571 3.464286 3.378571 14610   3    12
## 58440: 0.3928571 3.464286 3.378571 14610   3    12

# Get quantiles for extreme hot and cold of average daily temperature
q025 <- quantile(daily_data$m$tavg, 0.025)
q975 <- quantile(daily_data$m$tavg, 0.975)

The object daily_data is a list containing two datasets for females (f) and males (m). Each dataset contains:

• datedec: the date,
• age_bk: age buckets,
• nb_deaths: number of deaths,
• years: years,
• tmax: maximum temperature of the day
• tmax3: maximum temperature over the last 3 days
• tmin: minimum temperature over the last 3 days
• tmin3: minimum temperature over the last 3 days
• tavg: average temperature of the day
• tavg3: average temperature over the last 3 days
• time: day number as a integer number
• dow: day of the week as a integer number
• month: month number

Our DLNM model for nb_deaths is fitted based on the tavg variable, as well as datedec, dow and month.

2.2 Annual mortality data from the HMD

The annual mortality data considered are extracted from the Human Mortality Database . We downloaded the exposure to risk and death counts from the HMD website. We consider data from Metropolitan France for the calibration period \(\mathcal{T}_y = \{1980, \ldots, 2019\}\) and the age range \(\mathcal{X} = \{0, \ldots, 105\}\).

mort_dt <- get(load(file =  paste0(fold_data, "/mort_dt.RData")))

2.3 Data from climate models

We import daily data series for climate trajectories. The models selected in the paper are the 12 models available through the portal for RCP2.6, RCP4.5, and RCP8.5 scenarios. In this Notebook, we only consider the model CNRM-CM5 / ALADIN63 with the RCP8.5 scenario.

clim_model <- get(load(file = paste0(fold_data, 
                                     "ALADIN63_CNRM-CM5_RCP8.5.RData")))

3.1 Parameters

# Segmentation parameters
list_sexe <- list(m = 1, f = 2)
age_breaks <- c(0, 64, 74, 84, Inf)
age_labels <- c("0-64", "65-74", "75-84", "85+")

# Training period
annee_deb <- 1980
annee_fin <- 2019

# DLNM Parameters
param_dlnm_whole <- list(
  # main model, cubic natural spline with three internal knots in
  # the 10th, 75th, 90th percentiles of the temperature distribution
  varfun = "bs",
  vardegree = 2,
  varper  = c(10, 75, 90),
  ## specification of the lag function
  # Definition of the maximum lag, that is, 21 days
  lag  = 21,
  lagnk  = 3,
  ## degree of freedom for seasonality
  dfseas  = 8,
  ## degree of freedom for trend
  dftrend = NULL
  )

3.2 Model estimation

We fit the model for each sex and age bucket.

fit <- lapply(daily_data, function(x)
{
  fit_dlnm(x, param_dlnm_whole, per_age = T, summer = F, psi = NULL)
}
)

3.3 Results and diagnostics

We calculate the historical count of deaths attributed to temperatures for each day of the year \(d \in \mathcal{T}_d\), and age \(x \in X_k\), \(k \in \{1, \ldots, K\}\) using the formula \[ \bar{D}_{x,t,d}^{(g)}= (1-\exp{\left(- s(\vartheta_{d,t} ; L, \widehat{\boldsymbol{\theta}}_k ) \right)}) \times D_{x,t,d}^{(g)}. \] The term \(\exp{\left( s(\vartheta_{d,t} ; L, \widehat{\boldsymbol{\theta}}_k ) \right)}\) corresponds to the RR of temperature mortality. We display below the RR curves for temperatures.

summary_dlnm_all_sex(fit)

We then analyze the goodness of fit performance of our models. For this, we display the residuals of the Deviance by year and age group, and compare the distribution of observed (in blue) and predicted (in green) mortality counts for each month, distinguishing by sex and decade.

# in-sample performance of the fitted model for the whole year.
perf_mort <- lapply(names(daily_data), function(x){
  perf_dnlm(fit[[x]])
})
names(perf_mort) <- names(daily_data)

3.4 Estimation of the temperature- attributable fractions

The model’s setup allows for a range of predictions that precisely gauge the influence of heat, cold, or specific events occurring on each days of subset \(D \subseteq D^{\star}\).

The sum of the contributions from each day of this subperiod \(D\) for a year \(t \in \mathcal{T}_y\) is as follow \[ \bar{D}_{x,t}^{(g)} = \sum_{d \in D}{\bar{D}_{x,t,d}^{(g)}\mathbf{1}_{\left\lbrace d \in D\right\rbrace }}. \]

Thus, the total attributable fraction is estimated for each age \(x \in X_k\) and year \(t \in \mathcal{T}_y\) as \[ \theta_{x,t}^{(g)}= \dfrac{\bar{D}_{x,t}^{(g)}}{D_{x,t}^{(g)}}. \]

We compute the temperature-attributable fractions \(\theta^{(g)}_{x,t}\) for both women and men, and aggregate them over all ages to facilitate visualization on the figure below. The estimation error is calculated based on 20 replications of parametric bootstraps.

# Run excess of mortality
xs_mort <- lapply(names(daily_data), function(x){
  res <- excess_mort_dlnm(daily_data[[x]], fit[[x]], q_range = c(q025, q975),
                          nsim = nsim, parallel = parallel, ncpus = ncpus)
  return(res)
  })
names(xs_mort) <- names(daily_data)

# arrange data
xs_data <- lapply(names(xs_mort), function(i){
  xs_mort[[i]]$excess_mort %>%
    dplyr::mutate(gender = i) %>%
    rename(year = years)
})
xs_data <- do.call("rbind", xs_data)

# print attributable fractions
summary_temp_effect(xs_data)

4.1 Fitting the Poisson regression model

For calibrating the parameters \(({A}_x, {B}_x, {K}_{t}, {\alpha}_{x}^{(f)}, {\beta}_{x}^{(f)}, {\kappa}_{t}^{(f)}, {\alpha}_{x}^{(m)}, {\beta}_{x}^{(m)}, {\kappa}_{t}^{(m)})\), we use a Poisson assumption for the number of virtual deaths as in Brouhns, Denuit, and Vermunt (2002)

\[ \widetilde{D}_{x,t}^{(g)} \sim \text{Pois}\left( E_{x,t}^{(g)} \exp{\left( \widetilde{m}_{x,t}^{(g)}\right) }\right). \]
Thus, with the attributable fraction \(\theta^{(g)}_{x,t}\), we also have a Poisson formulation with log-link function for the overall number of deaths \[ {D}_{x,t}^{(g)} \sim \text{Pois}\left( E_{x,t}^{(g)} T_{x,t}^{(g)} \exp{\left( \widetilde{m}_{x,t}^{(g)}\right)}\right), \] where \(T_{x,t}^ {(g)} = \exp{\left( \bar{m}_{x,t}^{(g)}\right)} = 1 + \theta^{(g)}_{x,t}\), can be seen as an offset term.

xv <- 0:105
yv <- 1980:2019
countries <- c("FRATNP")
group <- c("Female", "Male", "Total")

# Data for women and men
xs_data <- get(load(file =  paste0(fold_data, "/xs_data.RData")))

xs_f <- xs_data %>% 
  dplyr::filter(gender == "f", temp_effect  == "all_effect") %>%
  dplyr::select(year, age_bk, attrib_frac)

xs_m <- xs_data %>% 
  dplyr::filter(gender == "m", temp_effect  == "all_effect") %>%
  dplyr::select(year, age_bk, attrib_frac)

# Transformation to matrix
transf_attrib_matrix <- function(df){
  mat <- matrix(data = 1, ncol = length(xv), nrow = length(yv))
  for(tt in yv){
    mat[tt - yv[1] + 1, 1:65] <- dplyr::filter(df, year == tt,
                                               age_bk == "0-64")$attrib_frac
    mat[tt - yv[1] + 1, 66:75] <- dplyr::filter(df, year == tt,
                                                age_bk == "65-74")$attrib_frac
    mat[tt - yv[1] + 1, 76:85] <- dplyr::filter(df, year == tt,
                                                age_bk == "75-84")$attrib_frac
    mat[tt - yv[1] + 1, 86:(max(xv) + 1)] <- dplyr::filter(
      df, year == tt, age_bk == "85+")$attrib_frac
  }
dimnames(mat) <- list(yv, xv)
return(mat)
}

xs_f <- transf_attrib_matrix(xs_f)
xs_m <- transf_attrib_matrix(xs_m)

# Create datasets for deaths with ajusted exposure
adj_mort_dt <- list(
  UNI = list(
    f = list(dtx = round(mort_dt$UNI$f$dtx),
             etx = mort_dt$UNI$f$etx * (1 + xs_f),
             wa = mort_dt$UNI$f$wa),
    m = list(dtx = round(mort_dt$UNI$m$dtx),
             etx = mort_dt$UNI$m$etx * (1 + xs_m),
             wa = mort_dt$UNI$m$wa)),
  ALL = list(dtx = mort_dt$ALL$dtx,
             etx = mort_dt$ALL$etx,
             wa = mort_dt$ALL$wa)
)
## replace exposure to adjusted exposure
adj_mort_dt$ALL$etx <- adj_mort_dt$UNI$f$etx + adj_mort_dt$UNI$m$etx

# Models with observed exposure to risk
fit_M <- fit_li_lee(xv, yv, yv, "m", data = mort_dt, method = "NR",
                    Ax = TRUE, exclude_coh = FALSE)
fit_F <- fit_li_lee(xv, yv, yv, "f", data = mort_dt, method = "NR",
                    Ax = TRUE, exclude_coh = FALSE)

# Models with adjusted exposure to risk
adj_fit_M <- fit_li_lee(xv, yv, yv, "m", data = adj_mort_dt, method = "NR",
                        Ax = TRUE, exclude_coh = FALSE)
adj_fit_F <- fit_li_lee(xv, yv, yv, "f", data = adj_mort_dt, method = "NR",
                        Ax = TRUE, exclude_coh = FALSE)

list_fit <- list(fit_F = fit_F,
                 fit_M = fit_M,
                 virt_fit_F = adj_fit_F,
                 virt_fit_M = adj_fit_M)
# print figures
myplot_parameters_li_lee(xv, yv, list_fit)

fig_res_f <- plot_residuals_li_lee(xv, yv, adj_fit_F, "FR", "Female")
fig_res_m <- plot_residuals_li_lee(xv, yv, adj_fit_M, "FR", "Male")

plot_grid(fig_res_f + ggtitle("Females"), 
          fig_res_m + ggtitle('Males'),
          label_size = 12)

4.2 Calibrating and forecasting the time series model

In this section, we focus on calibrating the time series model, rewritten in matrix form as \[ \boldsymbol{Y}_{t} = \boldsymbol{D}+ \boldsymbol{\Phi}\boldsymbol{Y}_{t-1} + \boldsymbol{E}_t, \] where \[ \boldsymbol{Y}_{t} = \begin{pmatrix} K_t \\ \kappa_t^{(f)} \\ \kappa_t^{(m)} \end{pmatrix}, \boldsymbol{D}= \begin{pmatrix} \delta \\ c^{(f)} \\ c^{(m)} \end{pmatrix}, \boldsymbol{\Phi}= \begin{pmatrix} 1 & 0 & 0\\ 0 & \phi^{(f)} & 0\\ 0 & 0 & \phi^{(m)}\\ \end{pmatrix} \text{ and } \boldsymbol{E}_t = \begin{pmatrix} e_t \\ r_t^{(f)} \\ r_t^{(m)} \end{pmatrix}. \]

We calibrate this model using two sets of parameters for \(\boldsymbol{Y}_{t}\), i.e.  the parameters obtained with the observed risk exposures and those obtained with the risk exposures adjusted for the effect of temperatures. These parameters are also estimated through maximum likelihood using R-package MultiMoMo.

## Forecasting Parameters
arima_spec <- list(K.t_M = "RWD", k.t_M = "AR1.1", k.t_F = "AR1.1")
n_ahead    <- length(start_year:end_year)
n_sim      <- nsim
est_method <- "PORT"

set.seed(1234)
# Model with original exposures
proj_par <- project_parameters(fit_M, fit_F, n_ahead, n_sim,
                               arima_spec, est_method)
# Model with adjusted exposures
adj_proj_par <- project_parameters(adj_fit_M, adj_fit_F, n_ahead, n_sim,
                                   arima_spec, est_method)

tab_param <- data.frame(
  v1 = c("Without temp. effects", "With temp. effects"),
  v2 = c(proj_par$coef_KtM, adj_proj_par$coef_KtM),
  v3 = c(proj_par$coef_ktM[1], adj_proj_par$coef_ktM[1]),
  v4 = c(proj_par$coef_ktM[2], adj_proj_par$coef_ktM[2]),
  v5 = c(proj_par$coef_ktF[1], adj_proj_par$coef_ktF[1]),
  v6 = c(proj_par$coef_ktF[2], adj_proj_par$coef_ktF[2])
)
# Print table
tab_param %>%
  kbl( booktabs = T,
       align='c',
       col.names = c("Calibration data", "$\\delta$",
                     "$c^{(m)}$", "$\\phi^{(m)}$",
                     "$c^{(f)}$", "$\\phi^{(f)}$"),
       digits=4, 
       escape =  FALSE) %>%
  kable_classic_2(full_width = F) %>%
  kable_styling(latex_options = "hold_position",  font_size = 8)

# Create correction matrix
corr_function <- function(S)
  {
    D <- diag(sqrt(diag(S)))
    Dinv <- solve(D)
    R <- Dinv %*% S %*% Dinv
    dimnames(R) <- dimnames(S)
    R 
  }

# Show correction matrix in table
tab_corr <- cbind(data.frame(corr_function(proj_par$cov_mat)), 
                  data.frame(corr_function(adj_proj_par$cov_mat)))
names(tab_corr) <- paste0("v", 1:ncol(tab_corr))
row.names(tab_corr) <- NULL

# Print table
tab_corr %>%
  dplyr::mutate(v = c("$e_t$", "$r_t^{(m)}$", "$r_t^{(f)}$"),
                .before = "v1") %>%
  kbl(booktabs = T,
       align='c',
       col.names = c("", "$e_t$", "$r_t^{(m)}$", "$r_t^{(f)}$",
                     "$e_t$", "$r_t^{(m)}$", "$r_t^{(f)}$"),
      digits=4, 
      escape =  FALSE) %>%
  add_header_above(c(" " = 1, "Without temp. effects" = 3,
                     "With temp. effects" = 3)) %>%
  kable_classic_2(full_width = F) %>%
  kable_styling(latex_options = "hold_position",  font_size = 8)

p <- plot_kt_sim(proj_par, adj_proj_par, n_sim, yv[1], end_year)
print(p)

5.1 Projecting future mortality

Now, we simulate the total mortality rates both with and without temperature effects \[ q_{x,t}^{(g)} = 1 - \exp{\left(-\widehat{m}_{x,t}^{(g)}\right)}, \widetilde{q}_{x,t}^{(g)} = 1 - \exp{\left(-\widetilde{m}_{x,t}^{(g)}\right)}, \]

For each simulation, we also compute the number of years of life expectancy lost (or gained) due to temperatures \[ \Delta e_{x,t}^{(g)} = \int_{x}^{t_{\max}}{e^{- \int_{x}^{t}{\widetilde{\mu}_{x,u}^{(g)}du}}dt} - \int_{x}^{t_{\max}}{e^{- \int_{x}^{t}{\mu_{x,u}^{(g)}du}}dt} \approx \sum_{k=1}^{t_{\max}}{\left[ \prod_{j=0}^{k-1}{\left(1-\widetilde{q}_{x,j}^{(g)}\right)} - \prod_{j=0}^{k-1}{\left(1-q_{x,j}^{(g)}\right)} \right]}, \] with the force of mortality \(\mu_{x,t}^{(g)} = \widetilde{\mu}_{x,t}^{(g)} + \bar{\mu}_{x,t}^{(g)}\).

The first step is to simulate the total death rates derived from models calibrated on data, whether adjusted or not for deaths attributable to temperatures.

## project mortality rates considering historical temperature effect 
proj_rates <- project_mortality_rates(fit_M, fit_F, proj_par)

## project mortality rates considering without temperature effect 
adj_proj_rates <- project_mortality_rates(adj_fit_M, adj_fit_F, adj_proj_par)

# reformat data
format_rates <- function(rates)
{
  rates <- melt(rates)
  names(rates) <- c("age", "year", "sim", "Qxt")
  rates <- rates %>%
    mutate(age_bk = cut(age, age_breaks, age_labels, include.lowest = T))
}

proj_rates <- lapply(proj_rates, format_rates)
names(proj_rates) <- c("m", "f")

adj_proj_rates <- lapply(adj_proj_rates, format_rates)
names(adj_proj_rates) <- c("m", "f")

Next, we apply the projected additional mortality related to future temperature. Estimation errors related to the DLMN model is considered with 20 bootstrap simulations.

It is worth noting that it is also possible to simultaneously consider temperature simulations from multiple climate models to capture this source of uncertainty. However, we do not include this step in this Notebook to simplify the presentation. Implementing this approach is not difficult and is been discussed in the paper.

# Change names dates
clim_model <- clim_model %>% dplyr::rename(datedec = date)
# Select the forecasting period
sc <- dplyr::filter(clim_model, years %in% start_year:end_year)

start_time <- Sys.time()
# Run simulations for males and females
res <- lapply(names(adj_proj_rates), function(x)
{
  # Select population
  temp_dem <- adj_proj_rates[[x]] %>%
    filter(year %in% start_year:end_year)
  # Launch forecasting along the rcp scenario with simulations
  # and parallelization
  f_mort <- forecast_mort_dlnm(temp_dem, sc, fit[[x]],
                               q_range = c(q025, q975),
                               nsim = nsim, parallel = parallel,
                               ncpus = ncpus)
  # Add gender indexes
  f_mort <- lapply(f_mort, function(tt){
    if(is.null(tt))
    {
      return(tt)
    } else{
      return(
        tt %>% 
          dplyr::mutate(gender = x) %>%
          relocate(gender, temp_effect, .before =  sim) 
      )
    }
  })
  return(f_mort)
})

# Merge more than two lists with the same element name
res <- Reduce(function(...) Map("rbind", ...), res)

# Save and printtime after operation
end_time <- Sys.time()
time_diff <- end_time - start_time
print(paste("Run time:", as.numeric(time_diff, units = "mins"), "mins"))

## [1] "Run time: 0.653317999839783 mins"

5.2 Dislaying attributable fraction

In the following figures, we display the projected attributable fraction related to future temperatures in the selected climate scenario.

To facilitate visual analysis, we calculate an aggregated attributable fraction to temperatures using the distribution of deaths known at the end of 2019, as follows \[ \theta_{t}(D) = \sum_{g \in \mathcal{G}} { \sum_{x \in \mathcal{X}}{\theta_{x,t}^{(g)} (D) \frac{D_{x, 2019}^{(g)}}{D_{2019}} }}, \]

# extract the last year 2019, for calculate exposure to risk
expo <- do.call("rbind", lapply(mort_dt$UNI, function(dt){
  dt <- dt$dtx[nrow(dt$dtx),  ]
  return(data.frame(age = 0:105,
                    w = dt,
                    age_bk = c(rep("0-64", 65), rep("65-74", 10),
                               rep("75-84", 10), rep("85+", 21))
                    ))
}))
expo$gender <- c(rep("f", 106), rep("m", 106))
# Agregate expo per age bucket
W <- sum(expo$w)
expo <- expo %>%
  group_by(age_bk, gender) %>%
  summarise(w = sum(w) / W)

# Specify the RCP number
res$tab_excess$traj_clim <- "ALADIN63_CNRM-CM5" 
res$tab_excess$rcp <- "8.5" 
plot_aff <- summary_forecast_temp_effect(res$tab_excess, expo)

# Plot aggregate temperature attributed fraction 
plot_aff[[1]]

p.comb <- plot_aff[[2]] 

# Plot temperature attributed fraction for each age group
p.comb[[1]]

p.comb[[2]]

p.comb[[3]]

p.comb[[4]]

5.3 Assessing the loss in life expectancy

Finally, we assess the loss in life expectancy due to future temperatures.

res$tab_ex$traj_clim <- "ALADIN63_CNRM-CM5" 
res$tab_ex$rcp <- "8.5" 
plot_ev <- summary_forecast_ev_effect(res$tab_ex)

# Plot LE including temperature effects
plot_ev[[1]]

# Plot lost in LE including temperature effects
plot_ev[[2]]